In order to obtain sharp lower bound of the Hausdorff dimension of the projection, we need that both \(\Phi (\theta ^+_{t,\mathbf {i}^-,\mathbf {j}^-})\)  and \(\Phi (\mu ^+\times \nu ^+)\)  are exact-dimensional and have the same dimension. When the IFSs satisfy the OSC, this fact is relatively easier to prove (may still need some work), but when there are possible overlaps in the IFSs, this fact is highly non-trivial to prove. We shall use the methods developed in [1]}, [2]} to prove such a result (Theorem REF ).
